Question

30 POINTS!!! HELP PLEASE!

Answer 1

#1)  In the sketch, the school is located at coordinates (1, 1).  Going north (up) 4 blocks and east (right) 3 blocks to get home places it at (4, 5).  The distance formula is:
$d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$.  Using our coordinates we have:
$d=\sqrt{(4-1)^2+(5-1)^2} \\=\sqrt{3^2+4^2} \\=\sqrt{9+16} \\=\sqrt{25} \\=5$  His house is 5 blocks from the school in a straight line.
#2)  The ladder (14 ft) forms the hypotenuse of a right triangle, with the legs being the distance from the house (6 ft) and the height of the ladder (b ft). This gives us:
$6^2+b^2=14^2 \\36+b^2=196$
Subtract 36 from both sides:
$36+b^2-36=196-36 \\b^2=160$
Take the square root of both sides:
$\sqrt{b^2}=\sqrt{160} \\b=12.6$
One possibility for the height of the second story window is 13 feet, since it is longer than 12.6.
#3)  The distance from 1st base to 3rd base forms the hypotenuse of a right triangle, with each leg being equal to s, the side length of the square formed by the baseball diamond.  Using the Pythagorean theorem we have:
$s^2+s^2=d^2$, where d is the distance from 1st to 3rd.  Combining like terms gives us $2s^2=d^2$.  Taking the square root of both sides we have $\sqrt{2s^2}=\sqrt{d^2} \\s\sqrt{2}=d$ (this is due to the fact that square root cancels a squared variable, so s comes out of the radical sign).
#4)  The straight distance from the boat's original position to its new position forms the hypotenuse of a right triangle, with the legs being the distance west (600) and the distance south (800) it traveled.  Using the Pythagorean theorem we have:
$600^2+800^2=d^2 \\360000+640000=d^2 \\1000000=d^2$
Taking the square root of both sides we have
$\sqrt{1000000}=\sqrt{d^2} \\1000=d$
The straight distance between the two points would be 1000 meters.